ON THE MOTION OF PENDULUMS. 29
lent V^"(r), and then integrating. It is unnecessary to add an arbitrary function of the time, since any such function may be supposed to be included in IT. We get
16. The integration of the differential equation (33) does not present the least difficulty, and (34) comes under a well-known integrable form. The integrals of these equations are
!rx .ixl- ............(38)>
^ = (7e~W V -mr/- V mr,
and we have to determine J., B, 0, D by the equations of condition.
The solution of the problem, in the case in which the fluid is confined by a spherical envelope, will of course contain as a particular case that in which the fluid is unlimited, to obtain the results belonging to which it will be sufficient to put 6 = GO . As, however, the case of an unlimited fluid is at the same time simpler and more interesting than the general case, it will be proper to consider it separately.
Let + m denote that square root of fju'~l n J — 1 which has its real part positive; then in equations (38) we must have D = 0, since otherwise the velocity would be infinite at an infinite distance. We must also have .5 = 0, since otherwise the velocity would be finite when r = co, as appears from (26). We get then from the equations of condition (35)
A = ia?c + ^(l + -1) 0 = -— (T 2iTfi \ ftnaj 2/fn
(/ ^ *} \ n
• 2/31/1 i " i O \ to
sin cN H-------1—2"o -
(V ma . maJ r
- — fl + — )e-"'(r-0'^ ..................(40),
ma \ mr/ '
p = ipoc/*'m« fl + — +4^) e-^'cos 6 ^ r r \ ma ma J r